Smectic Films and Focal Conic Domains

When smectic films are prepared on silicon wafers which are treated to induce a random planar anchoring of the smectic liquid crystal, the films are subject to antagonistic boundary conditions concerning the orientation of the smectic layers at the smectic/substrate and the smectic/air interface: The molecular smectic layers prefer to be parallel to the air interface but are forced to stand perpendicular on the substrate surface. The antagonistic anchoring conditions result in the formation of focal conic domains (FCDs): defect structures, in which the smectic layers are bend around two singular lines, namely a circle on the substrate surface and a straight line running from the circle center to the air interface.

Fig. 1: Left: schematic cross section through a row of focal conic domains and top view on the substrate plane; the blue ellipses symbolize the liquid-crystal molecules and the curved black lines the smectic layer planes. Right: dimensions of a single focal conic domain; diameter2r and film thickness H are typically between some μm and a few tens of μm, the depth h of the depressions in the air interface can amount up to one or two μm.

Fig. 1: Left: schematic cross section through a row of focal conic domains and top view on the substrate plane; the blue ellipses symbolize the liquid-crystal molecules and the curved black lines the smectic layer planes. Right: dimensions of a single focal conic domain; diameter2r and film thickness H are typically between some μm and a few tens of μm, the depth h of the depressions in the air interface can amount up to one or two μm.

Fig. 2: Cross section through two toroidal focal conic domains. The two singular lines (drawn in red) adopt for toroidal FCDs the shape of a circle and a straight line (in general, they form an ellipse and a hyperbola intersecting each others focal point).

Fig. 2: Cross section through two toroidal focal conic domains. The two singular lines (drawn in red) adopt for toroidal FCDs the shape of a circle and a straight line (in general, they form an ellipse and a hyperbola intersecting each others focal point).

In the μm thick films, FCDs arrange themselves in a regular lattice (see Fig. 3, top). AFM studies (see Fig. 3, bottom) reveal a depression in the film/air interface above each FCD, confirming the structure schemes shown in Figs. 1 and 2.

Fig. 3: Top: Optical micrograph of a lattice of FCDs in a smectic film on a silicon wafer inducing randam planar anchoring of the liquid crystal. Bottom: AFM image obtained for the same sample, the depth of the defect-induced depressions amounts to 250 nm, the lateral dimension of the shown area is 25 μm.

Fig. 3: Top: Optical micrograph of a lattice of FCDs in a smectic film on a silicon wafer inducing randam planar anchoring of the liquid crystal. Bottom: AFM image obtained for the same sample, the depth of the defect-induced depressions amounts to 250 nm, the lateral dimension of the shown area is 25 μm.

FCDs might be used as templates or matrices for self-organizing soft matter systems and our current studies are concerned with the controlled generation and arrangement of FCDs in smectic films.

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AFM Study of Defect-Induced Depressions of the Smectic-A/Air Interface
V. Designolle, S. Herminghaus, T. Pfohl, and Ch. Bahr, Langmuir 22, 363 (2006).
DOI: 10.1021/la0525224

Fig. 4: AFM image of a smectic film on a patterned substrate. Random planar anchoring is induced on 15 μm wide lines of a square-like grid pattern. The generation of focal conic domains (indicated by the surface depressions) is confined to these lines.

Fig. 4: AFM image of a smectic film on a patterned substrate. Random planar anchoring is induced on 15 μm wide lines of a square-like grid pattern. The generation of focal conic domains (indicated by the surface depressions) is confined to these lines.

Positioning FCDs by substrate patterning

A necessary condition for the formation of FCDs are the antagonistic anchoring conditions at the film surfaces. FCDs form only on substrates possessing random planar anchoring. If the substrate surface is modified such that some areas possess random planar anchoring and some areas homeotropic anchoring, the formation of FCDs is confined to the random planar anchoring areas. In practice, an anchoring pattern can be generated by evaporating a thin gold layer through an appropriate mask onto a random planar anchoring substrate. On the masked areas, the planar anchoring persists whereas the gold-coated areas possess homeotropic anchoring conditions.

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Controlling Smectic Focal Conic Domains by Substrate Patterning
W. Guo, S. Herminghaus, and Ch. Bahr, Langmuir 24, 8174 (2008).
DOI: 10.1021/la703717k

Fig. 5: Relation between the diameter 2r of the FCDs and the thickness H of the smectic film. The two data sets are obtained for two substrates possessing different strengths of the random planar anchoring. The anchoring strength was tuned by coating the silicon wafers with differently composed silane mixtures. Solid lines are fits to a theoretical model [J. B. Fournier, I. Dozov, G. Durand, Phys. Rev. A 41, 2252 (1990)].

Fig. 5: Relation between the diameter 2r of the FCDs and the thickness H of the smectic film. The two data sets are obtained for two substrates possessing different strengths of the random planar anchoring. The anchoring strength was tuned by coating the silicon wafers with differently composed silane mixtures. Solid lines are fits to a theoretical model [J. B. Fournier, I. Dozov, G. Durand, Phys. Rev. A 41, 2252 (1990)].

Controlling the size of FCDs by varying the anchoring strength

The lateral diameter of FCDs is determined by the thickness of the smectic film and the magnitude of the planar anchoring strength of the substrate. For a given substrate, possessing a certain anchoring strength, the diameter increases almost linearly with the film thickness, provided the thickness is larger than a certain critical value. Decreasing the anchoring strength decreases the slope of the diameter vs. film thickness relation and increases the value of the critical film thickness. The anchoring strength can be varied by coating the substrate with binary mixtures of silane compounds which induce, as pure agents, homeotropic or planar anhoring, respectively.

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Influence of anchoring strength on focal conic domains in smectic films
W. Guo and Ch. Bahr, Phys. Rev. E 79, 011707 (2009).
DOI: 10.1103/PhysRevE.79.011707

Fig. 6: Left: Relation between film thickness H, FCD diameter 2r, and depth h0 of the surface depression for strictly incompressible layers. Right: Temperature dependence of the ratio h/h0 (h: measured depth of the FCD-induced surface depressions, h0: calculated values assuming incompressible layers) in 40 μm thick films of various liquid crystal compounds. The four upper data sets belong to compounds possessing a smectic-A–isotropic transition, the two lower data sets belong to compounds possessing a smectic-A–nematic transition.TA↔N/I designates the smectic-A–nematic or smectic-A–isotropic transition temperature (solid lines are just guides to the eyes). — Fig. 6: Left: Relation between film thickness H, FCD diameter 2r, and depth h₀ of the surface depression for strictly incompressible layers. Right: Temperature dependence of the ratio h/h₀ (h: measured depth of the FCD-induced surface depressions, h₀: calculated values assuming incompressible layers) in 40 μm thick films of various liquid crystal compounds. The four upper data sets belong to compounds possessing a smectic-A–isotropic transition, the two lower data sets belong to compounds possessing a smectic-A–nematic transition.T_A↔N/I designates the smectic-A–nematic or smectic-A–isotropic transition temperature (solid lines are just guides to the eyes).

Fig. 6: Left: Relation between film thickness H, FCD diameter 2r, and depth h₀ of the surface depression for strictly incompressible layers. Right: Temperature dependence of the ratio h/h₀ (h: measured depth of the FCD-induced surface depressions, h₀: calculated values assuming incompressible layers) in 40 μm thick films of various liquid crystal compounds. The four upper data sets belong to compounds possessing a smectic-A–isotropic transition, the two lower data sets belong to compounds possessing a smectic-A–nematic transition.T_A↔N/I designates the smectic-A–nematic or smectic-A–isotropic transition temperature (solid lines are just guides to the eyes).

Influence of phase sequence on FCDs

When the smectic layers are regarded as completely incompressible, the depth h of the depressions in the air interface should depend only on the diameter 2r of the FCD and the thickness H of the smectic film: h = H − (H² − r²)^0.5 ≡ h₀. Experimentally, one measures often h values which are considerably smaller than the "geometrical" value h₀. We studied various liquid crystal compounds with different phase sequences. Whereas compounds possessing a direct smectic to isotropic transition showed h values similar to h₀, compounds with a smectic to nematic transition show a pronounced decrease of h with increasing temperature. A possible reason for this behavior is a decrease of the elastic constant controlling the layer compressibility at the second-order smectic - nematic transition.

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Influence of phase sequence on focal conic domains in smectic films
W. Guo and Ch. Bahr, Phys. Rev. E 79, 061701 (2009).
DOI: 10.1103/PhysRevE.79.061701

Structures on unidirectional planar anchoring substrates

Circular FCDs form on substrates possessing random planar anchoring conditions on which a preferred in-plane direction does not exist. We study also smectic films on substrates with unidirectional planar anchoring, on which a preferred in-plane alignment direction exists. On such substrates, linear structures are formed. Especially in thicker films additional modulations of these structures appear. The following AFM images are intended to give an impression of the diversity of the generated structures, many details are still to clarify: